Composite structures are advantageous over metallic structures in a number of ways, including corrosion resistance, reduced weight, and fatigue performance. However, the modes in which damage initiation and progression or fatigue occur are complex and extremely difficult to model analytically in comparison to traditional metallic structures.
As a result, certifying the durability of composite structures relies heavily upon experimental fatigue tests. Since fatigue scatter in composites has been, in general, much higher than in metallic components, experimental fatigue tests must be conducted on many replicates to achieve the desired levels of reliability in comparison with the same tests when performed on metallic structures. Carrying out such a vast array of experimental tests dramatically increases the cost and time necessary to complete certification of parts, particularly in the aerospace industry. In an attempt to account for the effects of fatigue scatter and to reduce the time and cost associated with testing components containing composite structures, a number of statistically based fatigue testing approaches have been developed. One such methodology uses the Load Enhancement Factor (LEF) approach originally developed by the Naval Air Development Center (NADC) in conjunction with Northrop Corporation in the 1980's in a series of papers on computational methods for computing fatigue life and residual strength in composites and certification methodologies. The tests generally include coupon testing and component testing. A “coupon” is a simple specimen constructed to evaluate a specific property of a material and a “component” is generally a more complex structure made from this material.
The new LEF approach was developed to overcome the individual disadvantages of the Load Factor and Life Factor approaches presented in, for instance, Ratwani, M. M. and Kan H. P., “Development of Analytical Techniques for Predicting Compression Fatigue Life and Residual Strength of Composites,” NADC-82104-60, March 1982 and Sanger, K. B., “Certification Testing Methodology for Composite Structures,” Report No. NADC-86132-60, January 1986. The Load Factor and Life Factor approaches are limited as they require either a high experimental test load which may exceed the static strength of the test material or component or require exceedingly long test periods. The LEF method uses these factors in concert to achieve both reasonable test durations and test loads that are well below the static strength of the component. However, the traditional methodology for determining LEF has been misused, resulting in the consistent use of an LEF value of 1.15 to represent all composite materials.
This approach has continued to be used, unmodified, over the past two decades. However, composite materials and the technology to apply and build from them have advanced considerably in the past two decades. In retrospect, the approach of the traditional method of calculating LEF relies on many questionable assumptions, is difficult to comprehend, and has a number of significant limitations. This has resulted in a number of inconsistencies and inaccuracies in the manner in which Load Enhancement Factors have been applied to fatigue testing on the component level in the past.
The traditional LEF methodology possesses a number of limitations and drawbacks, which render it difficult to employ and reduce its accuracy and applicability. The LEF methodology was derived using a “typical” stress-life (S-N) curve as a foundation. The papers outlining the development of the traditional LEF methodology identify PM as static strength. However, the equations used to derive the traditional methodology do not make use of static strength, PM in the equations. Therefore, this approach does not account for scatter in static strength, an assumption that is never identified by the methodology, but is implicit in the development of the equations. As composites have a high degree of scatter in static strength, this is a significant shortcoming of the traditional LEF approach.
Whitehead, et al., entitled “Certification Testing Methodology for Composite Structures,” Vols. I and II, October 1986, (“Whitehead”), a paper setting forth the traditional LEF approach, proposed to compute the LEF through Joint Weibull Analysis, which can be utilized to compute the shape and scale parameters assuming a Weibull distribution by measuring scatter within groups, for instance groups of fatigue stress levels, as set forth in volume I of the paper entitled “Certification Testing Methodology for Composite Structures”. Coupons must be grouped into distinct stress levels with many replicates per level. The authors provided the following equation listed below and recited as Equation 10 on Page 12 of Whitehead Vol. I to compute the shape ({circumflex over (α)}) and scale ({circumflex over (β)}) parameters,
                                                        ∑                              i                =                1                            M                        ⁢                          (                                                                    ∑                                          j                      =                      1                                                              n                      i                                                        ⁢                                                            x                      ij                                              a                        ^                                                              ⁢                    ln                    ⁢                                                                                  ⁢                                          x                      ij                                                                                                            ∑                                          j                      =                      1                                                              n                      i                                                        ⁢                                      x                    ij                                          a                      ^                                                                                  )                                -                      M                          a              ^                                -                      [                                          ∑                                  i                  =                  1                                M                            ⁢                                                                    ∑                                          j                      =                      1                                                              n                      fi                                                        ⁢                                      ln                    ⁢                                                                                  ⁢                                          x                      ij                                                                                        n                  fi                                                      ]                          =        0                            (                              Vol            .                                                  ⁢            I                    ,                      NADC            ⁢                                                  ⁢                          Eq              .                                                          ⁢              10                                      )            where:ni (i=1, 2 . . . , M) is the number of data points in the ith group of datanfi (i=1, 2 . . . , M) is the number of failures in the ith group of data.This equation is defined in terms of {circumflex over (α)} so an iterative procedure must be used to arrive at a solution.
Once {circumflex over (α)} is computed, the other Weibull parameters (scale parameters) can be determined using the following equation listed below and recited as Equation 11 on Page 12 of Whitehead Vol. I,
                                          β            ^                    i                =                              [                                          1                                  n                  fi                                            ⁢                                                ∑                                      j                    =                    1                                                        n                    i                                                  ⁢                                  x                  ij                                      a                    ^                                                                        ]                                1            /                          a              ^                                                          (                              Vol            .                                                  ⁢            I                    ⁢                                          ⁢          NADC          ⁢                                          ⁢                      Eq            .                                                  ⁢            11                          )            
On the surface, both equations appear to be valid. However, if equation (Whitehead Vol. I, Eq. 10) is dissected and rearranged slightly, the result is the following,
                    a        ^            ⁡              [                                            ∑                              i                =                1                            M                        ⁢                          (                                                                    ∑                                          j                      =                      1                                                              n                      i                                                        ⁢                                                            x                      ij                                              a                        ^                                                              ⁢                    ln                    ⁢                                                                                  ⁢                                          x                      ij                                                                                                            ∑                                          j                      =                      1                                                              n                      i                                                        ⁢                                      x                    ij                                          a                      ^                                                                                  )                                -                      [                                          ∑                                  i                  =                  1                                M                            ⁢                                                                    ∑                                          j                      =                      1                                                              n                      fi                                                        ⁢                                      ln                    ⁢                                                                                  ⁢                                          x                      ij                                                                                        n                  fi                                                      ]                          ]              M    =  1
In this form, the denominator on the left side of the equation is M, the total number of groups—here groups of fatigue stress levels. Therefore, the left side of the equation is merely an average value of M groups. The traditional methodology using Joint Weibull analysis does not identify the implications of using this simple statistical average in estimating the parameters. However, this method is severely limited. By using an average, the equation is only valid for stress levels with the same number of tested and failed coupons. In addition, this equation implies that nfi, the number of failures for a given stress level, must be equal for all stress levels. This significantly reduces the applicability of the method.
An S-N curve is typically developed using a number of specimens that are tested over varying stress levels to examine the effect of stress level on fatigue life. The general pattern of stress-life relationship shows that as stress increases life decreases, or in other words, higher levels of stress reduce life expectancy in a component.
However, using the traditional LEF approach and a traditional S-N curve, as previously stated, typically few if any of the data points in this type of relationship may be utilized to determine an LEF with an accounting of scatter as there is non-identical stress levels being tested and, typically, an uneven number of tested and failed coupons exists. To account for scatter in fatigue life, the traditional method using Joint Weibull Analysis requires identical specimens be tested at multiple identical stress levels for the comparison.
Furthermore, although the traditional LEF method clearly states that environmental conditions, specimen geometry, and numerous other variables affect the Weibull shape parameters, the method still uses these modal values “for simplicity,” as stated on page 82 of Whitehead, Vol. I. In addition, the reference claims that these modal values are “lower than mean values and, therefore, represent conservative values.” While the traditional LEF method with Joint Weibull Analysis may be correct in stating that the modal values may be less than the mean values for this particular literature review, these values do not necessarily represent conservative values for all composite materials, environmental conditions, specimen configurations, and other unknown variables. This is a potentially unsafe assumption.
Using these shape parameters values, the traditional LEF without Joint Weibull Analysis method then substitutes them into the following equation listed below and recited as Equation 17 on Page 46 of Whitehead Vol. II:
                    F        =                              μΓ            ⁡                          (                                                                    α                    R                                    +                  1                                                  α                  R                                            )                                                          [                                                -                                      ln                    ⁡                                          (                      p                      )                                                                                                                                                          χ                        γ                        2                                            ⁡                                              (                                                  2                          ⁢                          n                                                )                                                              /                    2                                    ⁢                  n                                            ]                                      1              /                              α                R                                                                        (                  NADC          ⁢                                          ⁢                      Eq            .                                                  ⁢            17                          )            
This assumes particular modal shape parameters for fatigue life and for residual strength, a single component-level test for a duration of 1.5 lifetimes at the B-Basis level (95% confidence with 90% reliability), the resulting LEF value was computed at approximately 1.15. Despite the widespread use of this LEF value (1.15), little data exists which substantiates the LEF values that engineers have employed on component tests. However, it is frequently assumed from the publication of this initial NADC value that these values can be repeatedly used and have been used on a wide number of composite compositions and structures being tested. In numerous published cases, the computations used to arrive at the LEF, Life Factors, etc. are absent from the documentation. Since many of these cases invoke the use of an LEF of 1.15, it is reasonable to assume that the engineers simply used the shape parameters and LEF values provided in the NADC document without quantifying the scatter of their specific materials under investigation.
However, due to the widespread variability in manufacturing processes, advances in composites, variations and advances in laminate design, and marked increases in the complexity of theses designs, material types, loading configurations, and the like, the LEF value of 1.15 when used, though assumed initially to be conservative, may in fact be unconservative given the large quantity of unknown variables.
Yet another confusion in the NADC approach is that static strength was used in the analysis. Residual strength will usually exhibit much more variability since damage has been induced in the material being tested, unlike static strength, which represents a material in a pristine condition. Using the static strength to compute a “representative” shape parameter is contradictory since static strength is never used in the derivation of the traditional LEF equations.
To overcome some of these drawbacks when testing on the coupon-level, if data already exists in this form, namely, one replicate per stress level, an alternative approach called “Weibull Regression for LEF Determination” can be utilized. This alternative approach can be used to model the S-N relationship, and then to develop LEF under that model.
The shortcomings, assumptions, and omissions of the traditional LEF computation methodology are further perpetuated and compounded by the use of these values in dramatically different geometric configurations, environments, manufacturing methods, and similar variables for composite structures. For example, some materials, such as aramid (sold under the trade name KEVLAR), are prone to substantial fatigue scatter when exposed to environments saturated with moisture.
Furthermore, certain loading conditions are prone to more scatter than others are. For example, laminate composites manufacturing using thick braids or tapes with large tow sizes exhibit more scatter in compressive loading (due to local buckling effects) than in tension loading. None of these factors is accommodated by the traditional method of calculating LEF. This result is another limitation in LEF analyses derived by the traditional LEF methodology. The failure to develop LEF analyses that consider a number of environmental variables, scatter in static strength, and other variables commonly at play in composite structure loading is a significant drawback to current testing practices.
Based upon these weaknesses, the approach that the traditional LEF methods take in generalizing the Weibull shape parameters and the resulting LEF value of 1.15 is of questionable merit and difficult to accept given the increasing number of unaccounted variables. Accordingly, the shape parameters and corresponding values should not be generalized or applied to all composite materials and a more comprehensive testing methodology is needed.
A new methodology for approaching the computation of the LEF incorporating the characteristics of scatter, residual strength, geometry and environmental variables and scatter in residual strength and fatigue life is needed to reduce costs, increase safety, and increase reliability. Due to the inherent cost and extended duration of testing of composite and metallic structures on the component-level, it is often desired to accelerate testing while still maintaining the desired level of statistical reliability and confidence. By adjusting both the load levels and planned duration of a component-level fatigue test without altering the statistical reliability, both the time and cost of the test may be reduced. Additionally, greater accuracy and higher levels of safety need to be achieved by a more accurate method of calculating the LEF accounting for residual strength and variations in residual strength through scatter. Accordingly, it is desirable to provide a method and an apparatus executing the method of improved computation of Load Enhancement Factors that is capable of overcoming the disadvantages described herein at least to some extent.